A visual metaphor for the Price Equation (UPDATED)

Disclaimer: This post is about something I’ve been struggling with for a while, and I was so pleased when I finally figured it out that I decided to post on it. Plus, I haven’t posted in a while so I figured I’d better post something. Whether these motivations are likely to lead to a broadly-interesting post is of course questionable. I’ve tried to set the problem I was struggling with in a wider context; only you can judge whether I’ve succeeded.

There are some questions in ecology and evolution to which the correct answer is, or is thought to be, dependent on one’s point of view. That is, they are questions that don’t have, or aren’t thought to have, answers completely dictated by empirical data and the rules of logic. In particular, think of questions which involve partitioning some total effect into components attributable to different causes. These kinds of questions crop up all the time in ecology and evolution. Whatever effect we’re studying typically comprises the net outcome of multiple, non-mutually exclusive mechanisms. Famous examples include the debate over how to partition the effects of different ‘levels of selection’ in evolutionary biology (Okasha 2006), and the debate over how to partition the total sums of squares in an unbalanced ANOVA (Stewart-Oaten 1995). Recent examples, on which I’ve published in Oikos, include debates in biodiversity-ecosystem function research over how to partition ‘selection effects’ from ‘complementarity effects’ (Fox 2005, Fox and Rauch 2009 Oikos), and how to partition ‘species richness effects’ from ‘species composition effects’ on ecosystem function and community stability (Fox 2006, Fox 2010 Oikos). Much as 4=2+2 and 4=1+3 are different but equally true equations, alternative partitions of the same total effect often can be viewed as different but equally true ways of dividing the same pie.

In debates about the appropriate way to partition a total effect, the answers typically fall into two groups. ‘Realist’ answers claim that there is one correct or best partitioning, although there is often disagreement as to what that correct partitioning is. For instance, the levels of selection debate often has focused on which way of partitioning group vs. individual selection correctly captures the way selection ‘really’ works in nature.* In contrast, ‘pluralist’ answers claim that there is no one best partitioning, that each has its own advantages and disadvantages (see Kerr and Godfrey-Smith 2002 for a pluralist take on the levels of selection).

Pluralists have an edge over realists: better visual metaphors. Pluralists often suggest that arguing about the correct partitioning of, say, group vs. individual selection is like arguing about whether this drawing shows a duck or a rabbit:

A realist who thinks this is ‘really’ a drawing of a duck, or ‘really’ a drawing of a rabbit, clearly is missing something important about the drawing. Of course, one still has to make an argument as to why, say, debates about the levels of selection are like debates about what an ambiguous drawing ‘really’ illustrates. But the visuals clarify and dramatize the pluralist position.

Realists lack an equally compelling visual metaphor for their position. This is something I’ve struggled with in my work on the Price Equation. It’s hard to explain how the Price Equation can be a mathematical tautology, like 4=2+2, but yet provide us with empirical insight. After all, there are infinite ways to rearrange any equation–what makes one arrangement better than another? 4=2+2 isn’t any more true than 4=1+3. I can of course argue my case, but I haven’t had a way to illustrate it–until now.

If ambiguous figures like the duck-rabbit are the canonical illustration of pluralism, the canonical illustration of realism (at least, realism about partitions) is sliding puzzles:

A sliding puzzle comprises n-1 square tiles arranged on a square grid with n slots. The goal is to slide the tiles from slot to slot in order to rearrange them into the desired pattern. The most famous of these is the ‘15 puzzle‘, which challenged the player to arrange tiles numbered 1-15 in numerical order.

Alternative arrangements of the tiles of a sliding puzzle are like alternative partitionings of a total effect. They all add up correctly–the tiles always cover the same total area, and they always contain the same information, no matter how you arrange them. But there’s only one arrangement that makes visual sense, that ‘looks right’, and that’s the correct arrangement. Most of the alternative arrangements make no sense at all:

I’m really pleased with this visual metaphor for partitioning a total effect in the ‘right’ way. As metaphors go, I think it’s quite accurate. Indeed, I’ve been wondering how far the metaphor can be pushed. Are sliding puzzles a useful metaphor for realism in science more generally, the way Susan Haack suggests that crossword puzzles are? I’ll have to think about that more. In the meantime, if I’m ever invited to talk about the Price Equation again at least I’ll finally have some sort of visual.

*footnote: I can share an interesting anecdote about a levels of selection ‘realist’ who changed his mind. Sober and Lewontin 1982 is a famous argument that genic selectionism is mere ‘bookkeeping’, a mathematical fiction which adds up correctly but which misdescribes the empirical facts of causality. As a graduate student at Rutgers in the late 1990s I was interested in the levels of selection debate. So I was thrilled to get the opportunity to meet one-on-one with Dick Lewontin when the great man came to Rutgers to give a seminar. When I asked him about this paper, Lewontin revealed that he’d changed his mind and become a pluralist. He no longer thought there were any empirical facts which would identify the ‘true’ level at which selection operates. As far as I know, he never published his revised views and so I don’t think his change of mind is widely known. That someone who’s thought as long and hard about the levels of selection as Dick Lewontin could change his mind illustrates just how difficult the choice between realism and pluralism can be.

UPDATE: In highlighting the levels of selection debate, I should have given a shout-out to a terrific recent Oikos paper, Kokko and Heubel 2011. This paper pushes the levels of selection debate forward by grounding it in the reproductive biology of an unusual all-female fish which, although asexual, requires sperm from heterospecific males as a developmental trigger. Kokko and Heubel also have some original and unusually thoughtful comments on pluralistic vs. realistic interpretations of alternative models of multilevel selection. As they say, read the whole thing!

I wonder about the application of this idea to the “inherently complex” phenomena you talked about in the Chess endgame post – would that be like a puzzle with hundreds of tiles, and we don’t know what the picture is supposed to be at the end?

By the way, when I respond to posts on this blog, I always find myself having to remove phrases from my drafts like “but reality should only manifest in one way,” because I think it makes me sound like a crazy person. ;0

“…would that be like a puzzle with hundreds of tiles, and we don’t know what the picture is supposed to be at the end?”

I tend to think of inherently-complex phenomena as phenomena for which no higher-level understanding (e.g., nothing equivalent to the Price equation) is possible at all. So maybe the analogy is that we just have a bunch of pieces and can’t even assemble them into a sliding puzzle in the first place?

“I always find myself having to remove phrases from my drafts…because I think it makes me sound like a crazy person.”

I’m the only one on this blog who ever risks sounding like a crazy person. ;-) Well, me and Jim Bouldin. ;-) (just kidding Jim!)